Corrigendum to "On injective modules and support varieties for the small quantum group"

Partial flag varieties and preprojective algebras

Let L be a preprojective algebra of Dynkin type, and let G be the corresponding complex semisimple simply connected algebraic group. We study rigid modules in subcategories sub(Q) for Q an injective L-module, and we introduce a mutation operation between complete rigid modules in sub(Q). This yields cluster algebra structures on the coordinate rings of the partial flag varieties attached to G.

Lifting preprojective algebras to orders and categorifying partial flag varieties

We describe a categorification of the cluster algebra structure of multi-homogeneous coordinate rings of partial flag varieties of arbitrary Dynkin type using Cohen-Macaulay modules over orders. This completes the categorification of Geiss-Leclerc-Schröer by adding the missing coefficients. To achieve this, for an order $A$ and an idempotent $e \in A$, we introduce a subcategory $\operatorname{CM}\nolimits_e A$ of $\operatorname{CM}\nolimits A$ and study its properties. In particular, under some mild assumptions, we construct an equivalence of exact categories $(\operatorname{CM}\nolimits_e A)…

A cluster realization of $U_q(\mathfrak{sl_n})$ from quantum character varieties

We construct an injective algebra homomorphism of the quantum group $U_q(\mathfrak{sl}_{n+1})$ into a quantum cluster algebra $\mathbf{L}_n$ associated to the moduli space of framed $PGL_{n+1}$-local systems on a marked punctured disk. We obtain a description of the coproduct of $U_q(\mathfrak{sl}_{n+1})$ in terms of the corresponding quantum cluster algebra associated to the marked twice punctured disk, and express the action of the $R$-matrix in terms of a mapping class group element corresponding to the half-Dehn twist rotating one puncture about the other. As a consequence, we realize the …

Representation ring of Levi subgroups versus cohomology ring of flag varieties II

For any reductive group G and a parabolic subgroup P with its Levi subgroup L, the first author in [Ku2] introduced a ring homomorphism $ ξ^P_λ: Rep^\mathbb{C}_{λ-poly}(L) \to H^*(G/P, \mathbb{C})$, where $ Rep^\mathbb{C}_{λ-poly}(L)$ is a certain subring of the complexified representation ring of L (depending upon the choice of an irreducible representation $V(λ)$ of G with highest weight $λ$). In this paper we study this homomorphism for G=Sp(2n) and its maximal parabolic subgroups $P_{n-k}$ for any $1\leq k\leq n$ (with the choice of $V(λ) $ to be the defining representation $V(ω_1) $ in $\…

Support Varieties and stable categories for algebraic groups

We consider rational representations of a connected linear algebraic group $\mathbb G$ over a field $k$ of positive characteristic $p > 0$. We introduce a natural extension $M \mapsto Π(\mathbb G)_M$ to $\mathbb G$-modules of the $π$-point support theory for modules $M$ for a finite group scheme $G$ and show that this theory is essentially equivalent to the more "intrinsic" and "explicit" theory $M \mapsto \mathbb P\mathfrak C(\mathbb G)_M$ of supports for an algebraic group of exponential type, a theory which uses 1-parameter subgroups $\mathbb G_a \to \mathbb G$. We extend our support theory…

Cohomology for Drinfeld doubles of some infinitesimal group schemes

Consider a field k of characteristic p > 0, G_r the r-th Frobenius kernel of a smooth algebraic group G, DG_r the Drinfeld double of G_r, and M a finite dimensional DG_r-module. We prove that the cohomology algebra H*(DG_r,k) is finitely generated and that H*(DG_r,M) is a finitely generated module over this cohomology algebra. We exhibit a finite map of algebras θ_r:H*(G_r,k) \otimes S(g) \to H*(DG_r,k) which offers an approach to support varieties for DG_r-modules. For many examples of interest, θ_r is injective and induces an isomorphism of associated reduced schemes. Additionally, for M an …

$\mathbb G$-模的滤过与增长

Filtrations and Growth of $\mathbb G$-modules

研究正特征域上仿射群概形的无限维模，通过上升收敛序列给出函子性滤过，刻画内射模，并利用有限维子余代数限制定义新不变量，探讨余有限模的增长。

Filtrations, 1-parameter Subgroups, and Rational Injectivity

We investigate rational $G$-modules $M$ for a linear algebraic group $G$ over an algebraically closed field $k$ of characteristic $p > 0$ using filtrations by sub-coalgebras of the coordinate algebra $k[G]$ of $G$. Even in the special case of the additive group $\mathbb G_a$, interesting structures and examples are revealed. The "degree" filtration we consider for unipotent algebraic groups leads to a "filtration by exponential degree" applicable to rational $G$ modules for any linear algebraic group $G$ of exponential type; this filtration is defined in terms of 1-parameter subgroups and is r…

组件表的魔力与奥秘

The Magic and Mystery of Component Tableaux

本文针对SL(n)情形系统研究了零纤维的不可约组件，通过构造半标准表与排除根向量得到子代数及Weierstrass截面，证明所得集合为不可约组件并建立单射的组件映射。